Quant Interview Questions (Interest Rate Strat.)

1)
Suppose you have a fair coin. How do you create an event A such that Pr(A) = 1/3?

Ans:
Toss the coin twice to produce the following space: {HH, HT, TH, TT}. Discard one of the element in the set. Each of the remaining outcome represents an event A.

Note: This recipe can be extended to producing an event with probability m/n, m and n being integers.

2)
Suppose you have a fair coin. How do you create an event A such that Pr(A) = r, r being an irrational number between 0 and 1?

Ans:
Translate r into a binary string. For example, 0.27314... would be {10}{0111}{011}{01}{100}... We start tossing the coin until we can tell if the string formed by coin tossing is greater/less than r in binary, with H = 1 and T = 0. For example, the first step is to toss twice. If we have {TH} = {01}, then the string formed by coin tossing must be less than r in binary. We terminate and conclude that the event belongs to A; If we have {HH} = {11}, then the string formed by coin tossing must be greater than r in binary. We terminate and conclude that the event does not belong to A. If the two tosses produce {HT} = {10} then we cannot conclude, and have to move on to the next digit {111}, and so on. (MC verification)

Note: We have to append a zero in front of {111}, {11} and {1} because otherwise we cannot create a greater binary, i.e. 111 is the greatest 3-digit binary number.

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